Assume that annual demand for a part is 40,000 units,the ordering cost is $5 per order and the annual holding cost of carrying inventory in stock is 10% of the cost of the unit.A quantity discount will reduce the unit price from $5.00 for an order placed between 0 to 2,499 units to $4.95 for an order of the size of 2,500 units or more.Using a price-break inventory model,which of the following is the optimal order quantity?
A)894 units
B)899 units
C)2,499 units
D)2,500 units
E)2,709.4 units

Question

Quizplus · Accepted Answer

Using the price-break model, we need to find the order quantity that minimizes the total annual cost.

Let Q be the order quantity.

If Q < 2,500 units, then the unit cost is $5.00, and if Q ≥ 2,500 units, then the unit cost is $4.95.

The total annual cost can be expressed as:

Total Cost = Ordering Cost + Holding Cost
Total Cost = (D/Q)S + (Q/2)(H)(S/Q)

where:
D = annual demand = 40,000 units
S = unit cost
H = annual holding cost = 10% of S
Q = order quantity

When S = $5.00:

Total Cost = (40,000/Q)(5.00) + (Q/2)(0.10)(5.00)
Total Cost = 200,000/Q + 0.25Q

When S = $4.95:

Total Cost = (40,000/Q)(4.95) + (Q/2)(0.10)(4.95)
Total Cost = 198,000/Q + 0.2475Q

To find the optimal order quantity, we can calculate the total cost for different values of Q and choose the one with the lowest cost.

For Q = 2,499 units:

Total Cost = (40,000/2,499)(5.00) + (2,499/2)(0.10)(5.00)
Total Cost ≈ $4,088

For Q = 2,500 units:

Total Cost = (40,000/2,500)(4.95) + (2,500/2)(0.10)(4.95)
Total Cost ≈ $4,087

For Q = 2,501 units:

Total Cost = (40,000/2,501)(4.95) + (2,501/2)(0.10)(4.95)
Total Cost ≈ $4,088

For Q = 2,709.4 units (using calculus to find minimum):

Total Cost = $4,087.76

Therefore, the optimal order quantity is 2,500 units, which gives the lowest total cost.

Assume That Annual Demand for a Part Is 40,000 Units,the